Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

De-Equivariantization of Hopf Algebras

We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras A(H, G, Φ) attached to a coradically-graded pointed Hopf algebra H and some extra data.     

Character tables and normal left coideal subalgebras

We continue studying properties of semisimple Hopf algebras H over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of H reflects normal left coideal subalgebras of H. These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze ‘semi-kernels’ and their relations to the character table. We prove a full analogue of the Burnside–Brauer theorem for almost cocommutative H. We also prove the Hopf algebras analogue of the following (Burnside) theorem: If G is a non-abelian simple group then {1} is the only conjugacy class of G which has prime power order.     

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$algebras

Let G be a group and assume that (A _p ) _p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ _p,q : A _pq → A _p ⊗ A _q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ _p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.     

Hopf Algebras and the Penrose Polynomial

Let λ be a positive integer and let G be a plane graph. LetP (G, λ) be the Penrose polynomial of G. We will present an interpretation ofP (G, − λ) in terms of colourings of G. In order to prove our main theorem we construct a Hopf algebra A of graphs and a homomorphism of Hopf algebras Ψ fromA onto a Hopf algebra of polynomials in one indeterminate. If G is a plane graph, thenΨ (G) coincides with the Penrose polynomial of G.     

Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

In this paper, first we introduce a new notion of commuting condition that φφ ₁ A = A φ ₁ φ between the shape operator A and the structure tensors φ and φ ₁ for real hypersurfaces in G ₂(? ^m+2). Suprisingly, real hypersurfaces of type (A), that is, a tube over a totally geodesic G ₂(? ^m+1) in complex two plane Grassmannians G ₂(? ^m+2) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G ₂(? ^m+2) satisfying the commuting condition. Finally we get a characterization of Type (A) in terms of such commuting condition φφ ₁ A = A φ ₁ φ.     

A Class of Multiplier Hopf Algebras

We compute the Drinfel’d double for the bicrossproduct multiplier Hopf algebra A = k[G] ⋊ K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] ⋊ K(H) if there is a factor-reversing group isomorphism. Presented by A. Verschoren.     

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.     

Free products of kω-topological groups with normal amalgamation

It is shown in this paper that if A is a closed normal subgroup of k_ω-topological groups G and H, then the free product of G and H with A amalgamated, G1_AH, exists, is Hausdorff and indeed a k_ω-group.     

Finite quantum groups over abelian groups of prime exponent

We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.     

Picard Groups of Rings of Coinvariants

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C). Presented by: Claus Ringel.     

Diagrams of Hopf algebras with the Chevalley property

In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A’s diagram equaling the diagram of its maximal pointed Hopf subalgebra.     

Hopf powers and orders for some bismash products

The exponent of a finite group G can be viewed as a Hopf algebraic invariant of the group algebra H=kG: it is the least integer n for which the nth Hopf power endomorphism [n] of H is trivial. The exponent of a group scheme G as studied by Gabriel and Tate and Oort can be defined in the same way using the coordinate Hopf algebra H=O(G).The power map and the corresponding notion of exponent have been studied for a general finite-dimensional Hopf algebra beginning with work of Kashina. Several positive results, suggested by analogy to the group case, were proved by Kashina and by Etingof and Gelaki.Given these positive results, there was some hope that the Hopf order of an individual element of a Hopf algebra might also be a well-behaved notion, with some properties analogous to well-known facts on the orders of elements of a finite group.In fact we prove that such analogous properties do hold for Hopf algebras satisfying the usual rule for iterated powers; for example, such a Hopf algebra H has an element of order n if and only if n divides the exponent of H. However, in general such properties are not true. We will give examples where the behavior of Hopf powers, Hopf orders, and related notions is rather strange, unexpected, and seemingly hard to predict. We will see this using computer algebra calculations in Drinfeld doubles of finite groups, and more generally in bismash products constructed from factorizable groups.     

Graded Associative Conformal Algebras of Finite Type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group Γ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group G such that the identity component G ⁰ is the affine line and G/G ⁰???Γ. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.     

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

In this article we develop some of the basic constructions of the theory of Hopf algebras in the context of autonomous pseudomonoids in monoidal bicategories. We concentrate on the notion of Hopf modules. We study the existence and the internalisation of this notion, called the Hopf module construction. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case a Hopf module construction for A always exists. We recover from the general theory developed here results on coquasi-Hopf algebras.     

Homological dimension of weak Hopf�CGalois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra and B the subalgebra of the H-coinvariant elements of?A. Let A/B be a right weak H-Galois extension. We prove that A/B is a separable extension if H is semisimple. Using this, we show that the global dimension and weak dimension of A are less than those of?B. As an application, we obtain Maschke-type theorems for weak Hopf?CGalois extensions and weak smash products.     

The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a *-representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π:A(G)→B(H) is similar to a *-representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a *-representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].     

Crossed products of type I af algebras by abelian groups

Let (G, A, α) be a separableC*-dynamical system, withG abelian, and let Γ denote the dual group ofG. We characterize the Γ-invariant ideals of the crossed product algebraG×∩A, and use this characterization to prove that if in additionG is compact andA is type I AF, thenG×∩A is AF also. Finally, assumingG is discrete abelian and bothA andG×∩A are type I. we determine necessary and sufficient conditions, in terms ofA and the isotropy subgroups for the action ofG onÂ, forG×∩A to be AF.